Time-dependent non-Hermitian Hamiltonians Research Articles

On the η pseudo symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, . We propose a derivation of pseudo symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results.Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail

Non-Hermitian Floquet-free analytically solvable time-dependent systems [Invited

The non-Hermitian models, which are symmetric under parity (P) and time-reversal (T) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT-symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.

Entanglement between uncoupled modes with time-dependent complex frequencies

In this work we present the general unified description for the unitary time evolution generated by time-dependent non-Hermitian Hamiltonians embedding the bosonic representations of $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras. We take into account a time-dependent Hermitian Dyson maps written in terms of the elements of those algebras with the relation between non-Hermitian and its Hermitian counterpart being independent of the algebra realization. As a direct consequence, we verify that a time-evolved state of uncoupled modes modulated by a time-dependent complex frequency may exhibits a nonzero entanglement even when the cross operators, typical of the interaction between modes, are absent. This is due the nonlocal nature of the nontrivial dynamical Hilbert space metric encoded in the time-dependent parameters of the general Hermitian Dyson map, which depend on the imaginary part of the complex frequency. We illustrate our approach by setting the $\mathcal{PT}$-symmetric case where the imaginary part of frequency is linear on time for the two-mode bosonic realization of Lie algebras.

Non-Hermitian Hamiltonian beyond PT symmetry for time-dependent [formula omitted] and [formula omitted] systems — Exact solution and geometric phase in pseudo-invariant theory

In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.

Beyond PT-symmetry: Towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

In this work we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t)I(t) for a TD non-Hermitian Hamiltonian H(t)H(t). Assuming H(t)H(t) to be simultaneously \rho(t)ρ(t)-pseudo-Hermitian and \Xi(t)Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t)I(t)=Ξ−1(t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J. Math, Phys. 43, 3944 (2002)] for the time-independent (TI) scenario. We apply our method for the derivation of the symmetries associated with TD non-Hermitian linear and quadratic Hamiltonians. The computed TD symmetry operators for both cases are then particularized for their equivalent TI Hamiltonians and PT -symmetric restrictions. In the TI scenario we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator: I^{2k}=1I2k=1 with kk odd [J. Phys. A 35, L467 (2002)]. The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians. From this relation the TD metric is automatically derived from the TD symmetry of the problem. Then, when placed in perspective with the antilinear symmetry I(t)=\Xi^{-1}(t)\rho(t)I(t)=Ξ−1(t)ρ(t), the symmetry-metric relation finally allow us to derive the \Xi(t)Ξ(t)-anti-pseudo-Hermitian operator. Our results reinforce the prospects of going beyond \mathcal{PT}𝒫𝒯-symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.

Enhancement of photon creation through the pseudo-Hermitian Dynamical Casimir Effect

We analyse here the pseudo-Hermitian Dynamical Casimir Effect, proposing a non-Hermitian version of the effective Law’s Hamiltonian used to describe the phenomenon. We verify that the average number of created photons can be substantially increased, a result which calls the attention to the possibility of engineering the time-dependent non-Hermitian Hamiltonian we have assumed. Given the well-known difficulty in detecting the Casimir photon production, the present result reinforces the importance of pseudo-Hermitian quantum mechanics as a new chapter of quantum theory and an important tool for the amplification of Hermitian processes such as the degree of squeezing of quantum states.

Non-Hermitian Hamiltonian Beyond Pt -Symmetry for Time-Dependant Su(1,1) and Su(2) Systems --- Exact Solution and Geometric Phase in Pseudo-Invariant Theory

We investigate in this paper time-dependent non-Hermitian Hamiltonians, which consist respectively of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1)and SU(2)systems in a unified manner. Then, we derive the LR phase, which can be separated to the dynamic phase and the geometrical phase. The analytical results are exactly in agreement with those of corresponding Hermitian Hamiltonians in the literature.

A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians* *This paper is dedicated to the memory of our friends and colleagues Lahcene Alouani and Mebrouk Gherda died due to covid 19.

With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the −inner product, where is a obtained from the charge conjugation operator through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

Infinite series of time-dependent Dyson maps

We propose and explore a scheme that leads to an infinite series of time-dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying symmetries responsible for this feature respected by various Lewis–Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric operators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian -symmetrical fashion.

An attractor dynamics in a non-Hermitian two-level system

Exceptional points in non-Hermitian systems possess fascinating properties. We present an exactly solvable attractor dynamics for the first time from the two-level time-dependent non-Hermitian Hamiltonian. This allows the evolution from a pure or mixed initial state to the coalescence state by varying the imaginary parameter along a specific diabatic passage. In contrast to a chaotic attractor that is ultrasensitive to initial conditions, the designed attractor is insensitive to initial conditions. The attractor-like behavior is applicable to several adiabatic processes.

Dynamically Encircling an Exceptional Point in a Real Quantum System.

The exceptional point, known as the non-Hermitian degeneracy, has special topological structure, leading to various counterintuitive phenomena and novel applications, which are refreshing our cognition of quantum physics. One particularly intriguing behavior is the mode switch phenomenon induced by dynamically encircling an exceptional point in the parameter space. While these mode switches have been explored in classical systems, the experimental investigation in the quantum regime remains elusive due to the difficulty of constructing time-dependent non-Hermitian Hamiltonians in a real quantum system. Here we experimentally demonstrate dynamically encircling the exceptional point with a single nitrogen-vacancy center in diamond. The time-dependent non-Hermitian Hamiltonians are realized utilizing a dilation method. Both the asymmetric and symmetric mode switches have been observed. Our Letter reveals the topological structure of the exceptional point and paves the way to comprehensively explore the exotic properties of non-Hermitian Hamiltonians in the quantum regime.

Dynamical formulation of low-energy scattering in one dimension

The transfer matrix M of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical formulation of stationary scattering. We explore the utility of this formulation in the study of the low-energy behavior of the scattering data. In particular, for the exponentially decaying potentials, we devise a simple iterative scheme for computing terms of arbitrary order in the series expansion of M in powers of the wavenumber. The coefficients of this series are determined in terms of a pair of solutions of the zero-energy stationary Schrödinger equation. We introduce a transfer matrix for the latter equation, express it in terms of the time-evolution operator for an effective two-level quantum system, and use it to obtain a perturbative series expansion for the solutions of the zero-energy stationary Schrödinger equation. Our approach allows for identifying the zero-energy resonances for scattering potentials in both full line and half-line with zeros of the entries of the zero-energy transfer matrix of the potential or its trivial extension to the full line.

Perturbative approach for strong and weakly coupled time-dependent for non-Hermitian quantum systems

We propose a perturbative approach to determine the time-dependent Dyson map and the metric operator associated with time-dependent non-Hermitian Hamiltonians. We apply the method to a pair of explicitly time-dependent two dimensional harmonic oscillators that are weakly coupled to each other in a PT-symmetric fashion and to the strongly coupled explicitly time-dependent negative quartic anharmonic oscillator potential. We demonstrate that once the perturbative Ansatz is set up the coupled differential equations resulting order by order may be solved recursively in a constructive manner, thus bypassing the need for making any guess for the Dyson map or the metric operator. Exploring the ambiguities in the solutions of the order by order differential equations naturally leads to a whole set of inequivalent solutions for the Dyson maps and metric operators implying different physical behaviour as demonstrated for the expectation values of the time-dependent energy operator.

Time-dependent pseudo-bosonic coherent states

In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).

Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators

We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from below. All transformations are carried out non-perturbatively so that all Hamiltonians in this process are spectrally exactly equivalent in the sense that they have identical instantaneous energy eigenvalue spectra.

Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps

We present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of the whole time-dependent non-Hermitian Hamiltonian or parts of it. The time-dependent Dyson map is derived through a constructed Schrödinger-like equation governed, in one case, by the non-Hermitian Hamiltonian itself or, in another case, by its parts. In the former case, when the whole non-Hermitian Hamiltonian is considered, our scheme ensures the time-independence of the metric operator despite the time-dependence of the Dyson map, a necessary condition for the observability of the non-Hermitian Hamiltonian. In the later case, however, when parts of the non-Hermitian Hamiltonian is considered, our method ensures the simultaneous time-dependence of the Dyson map and the metric operator. In this latter case what is ensured is the observability of the remaining part of the non-Hermitian Hamiltonian that was not chosen for the derivation of the Dyson map. Illustrative examples, for both cases, are derived from a driven non-Hermitian Harmonic oscillator.

Ladder Invariants and Coherent States for Time-Dependent Non-Hermitian Hamiltonians

We extend the Dodonov–Malkin–Man’ko–Trifonov (DMMT) invariant method (Malkin et al. Phys. Rev. D 2, 1371 1, J. Math. Phys. 14, 576 2; Dodonov et al. Int. J. Theor. Phys. 14, 37 3; Dodonov and Man’ko Phys. Rev. A 20, 550 4) to time-dependent pseudo-fermionic systems by introducing ladder invariant operators (time-dependent integrals of motion) which play the role of time-dependent pseudo-fermionic operators and constructing time-dependent pseudo-fermionic coherent states (PFCS) for such systems. As illustrative example, we study in details the time-dependent parity-time-symmetric two-level system under synchronous combined modulations. We explicitly determine time-dependent pseudo-fermionic operators and construct time-dependent PFCS for this physical system. We show that our approach can be extended to time-dependent pseudo-bosonic systems.

Exceptional points in classical spin dynamics

Non-conservative physical systems admit a special kind of spectral degeneracy, known as exceptional point (EP), at which eigenvalues and eigenvectors of the corresponding non-Hermitian Hamiltonian coalesce. Dynamical parametric encircling of the EP can lead to non-adiabatic evolution associated with a state flip, a sharp transition between the resonant modes. Physical consequences of the dynamical encircling of EPs in open dissipative systems have been explored in optics and photonics. Building on the recent progress in understanding the parity-time ({mathscr{P}}{mathscr{T}})-symmetric dynamics in spin systems, we use topological properties of EPs to implement chiral non-reciprocal transmission of a spin through the material with non-uniform magnetization, like helical magnet. We consider an exemplary system, spin-torque-driven single spin described by the time-dependent non-Hermitian Hamiltonian. We show that encircling individual EPs in a parameter space results in non-reciprocal spin dynamics and find the range of optimal protocol parameters for high-efficiency asymmetric spin filter based on this effect. Our findings offer a platform for non-reciprocal spin devices for spintronics and magnonics.

Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.

Evolution operator for time-dependent non-Hermitian Hami ltonians

The evolution operator \(U(t)\) for a time-independent parity-time-symmetric systems is well studied in the literature. However, for the non-Hermitian time-dependent systems, a closed form expression for the evolution operator is not available. In this paper, we make use of a procedure, originally developed by A.R.P. Rau [Phys.Rev.Lett, 81, 4785-4789 (1998)], in the context of deriving the solution of Liuville-Bloch equations in the product form of exponential operators when time-dependent external elds are present, for the evaluation of \(U(t)\) in the interaction picture wherein the corresponding Hamiltonian is time-dependent and in general non-Hermitian. This amounts to a transformation of the whole scheme in terms of addressing a nonlinear Riccati equation the existence of whose solutions depends on the fulllment of a certain accompanying integrabilty condition.